Although this follows from Theorem B by [NXY19, Theorem 6.1] (end of the proof) and by [NXY19, §3.4], we sketch the proof for reader’s convenience.
As mentioned in Section 1.5, the retraction $\rho_{\mathscr{X}}$ over $\Star(\tau_{Z})$ only depends on $\widehat{\mathscr{X}_{/Z}}$, so that by Theorem B we may assume that $\mathscr{X}$ is a toric $R$-scheme. The equality $\rho_{\mathscr{X}}=\val$ now holds over $\Star(\tau_{Z})$ by Proposition 1.5.2, so that it follows from Definition 1.6.1 that $\rho_{\mathscr{X}}$ is an affinoid fibration over $\Star(\tau_{Z})$. ∎